PHYSICAL
PRL 114, 062006 (2015)
REVIEW
week ending 13 FEBRUARY 2015
LETTERS
Three Loop Cusp Anomalous Dimension in QCD $
Andrey Grozin Budker Institute o f Nuclear Physics SB RAS, Novosibirsk 630090, Russia and Novosibirsk State University, Novosibirsk 630090, Russia Johannes M. Henn' Institute for Advanced Study, Princeton, New Jersy 08540, USA Gregory P. Korchemsky* Institut de Physique Theorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France Peter Marquard§ Deutsches Elektronen-Synchrotron, DESY, Platanenallee 6, D15738 Zeuthen, Germany (Received 12 September 2014; published 13 February 2015) We present the full analytic result for the three loop angle-dependent cusp anomalous dimension in QCD. With this result, infrared divergences of planar scattering processes with massive particles can be predicted to that order. Moreover, we define a closely related quantity in terms of an effective coupling defined by the lightlike cusp anomalous dimension. We find evidence that this quantity is universal for any gauge theory and use this observation to predict the nonplanar ^-dependent terms of the four loop cusp anomalous dimension. DOI: 10.1103/PhysRevLett.ll4.062006
PACS numbers: 12.39.Hg, 11.15.-q
Understanding the structure of soft and collinear diver gences is of great theoretical interest in quantum field theory. It is also relevant for phenomenological applications such as the production of heavy particles at the LHC, where effects from soft gluon radiation need to be resummed in order to improve theoretical predictions. It is well known that the infrared (or long-distance) asymptotics of scattering amplitudes is described by correlation functions of Wilson lines pointing along the momenta of the scattered particles [1,2]. The latter satisfy evolution equations with the corresponding anomalous dimension being, in general, a matrix in color space. In the planar limit, this matrix is expressed in terms of the twoline cusp anomalous dimension [3]. The two loop result for this fundamental quantity has been known for more than 25 years [4]; see, also, Ref. [5], Here we report on the full result for the cusp anomalous dimension in QCD at three loops. To compute the cusp anomalous dimension, we consider the vacuum expectation value of the Wilson line operator W
N
,
= VyV2,
0031 -9 0 0 7 /1 5 /1 14(6)/062006(5)
x = e1^,
2cos = x + l / x .
(3)
In Euclidean space | jc| = 1, whereas for Minkowskian angles (p = id (with 6 real), the variable x can take arbitrary
( 1)
with A ^ x ) = A °(x )T a and T a being the generators of the fundamental representation of the SU (N ) gauge group. Here, the integration contour C is formed by two segments along directions v\ and (with v\ = v\ — Y), with (Euclidean space) cusp angle tp, COS (p
cf. Fig. 1. Perturbative corrections to the Wilson loop (1) contain both ultraviolet (cusp) and infrared divergences. We employ dimensional regularization with D = 4 - 2e to regularize the former and introduce an infrared cutoff using the heavy quark effective theory framework. The cusp anomalous dimension Tcusp is extracted as the residue at the simple pole l / e in the corresponding renormalization factor. It depends on the cusp angle 1
* {1) = Cp,
K® = CACF
(6)
The previously known one and two loop [4] results can be written as p (l)
_ C A
week ending 13 FEBRUARY 2015
LETTERS
'67
/r2\
5
36
12 )
9
'245
67 n2
96
216
+ C2FrifTp( £3 liusp
—
67 5 V — CfC4 - - C FTFn f \ A x.
x cFcA[a 3 + a 2] +
( 8)
At three loops, we find "cusp — c\CFC\ + c2Cf (TFnf y + CjCpT Fnf + c^CFCAT Ffij,
(9)
with 1
c\ — ^ [^5 +
a 4 + b 5 + b 3] +
67 29 — a 3 + Yg^2
'245 11 \ ~ > + 2 4 CV A l ’ 1 =~ 2 J A l ’
c4
( 55\ ^ 1, C3 = ( & - 4 8 )
209 n 5 r~ ~ , 1 /' - 9 [a 3 + a 2] - - ( % + l 6 ,K
( 10)
(ii)
- cusp (as,x)x= K
( a s) log(l/jc) + O(x0),
55
1
48/
27
Cp(nf TF)2,
(14)
rcusp).
Q.(a,x)—r cusp(as,x),
a ~ n / C FK(as),
(15)
where Tcusp and K(as) are evaluated in the same scheme (and for the same theory). By construction, Q has the universal limit S2(a,x)JC= 0- C f-l og(l /x) + O(x0),
n
(16)
as one can easily verify by comparing to Eq. (13). Using the results up to three loops given in Eqs. (7)-(9) and (14), and expanding both sides of the first relation in Eq. (15) to third order in as, we find
f a \ 2 CACF
£2(m, x) — —CpAx T [ — (« )
+
a \ 3CpC2A
A A ~A A3 + A2 + K—2 A t
2 f
A5 + A4 —A 2 + B5 +
nA ~ n2 ~ n2 ~ + y A 3 + T A2 - — Aj + 0 ( a 4).
(13)
with K(as) being the lightlike cusp anomalous dimension. To three loops, it is given by [17]
24'
where K(as) = ^2m>i{ocs/n)mK^m\ We found perfect agreement for all terms. Finally, if the conformal symmetry of (massless) QCD were not broken, one would expect that the cusp anomalous dimension should be related in the antiparallel lines limit (p — n - 8, 8 -* 0, to the quark-antiquark potential [18] (at one loop order lower compared to Starting from Eq. (9), we indeed find perfect agreement with the result quoted in the second of Ref. [19], up to conformal symmetry breaking terms proportional to the QCD /I function. Our result for the cusp anomalous dimension is valid in the MS (dimensional regularization) scheme. Going to the DR (dimensional reduction) scheme amounts to a finite renormalization of the coupling constant. We can introduce a quantity Q. which is the same in both schemes by switching from as to an “effective cou pling” a,
( 12)
Here, CF = (N2 - 1 ) /(22V) and CA = N are the quadratic Casimir operators of the SU(N) gauge group in the fundamental and adjoint representation, respectively, rif is the number of quark flavors, and TF = 1/ 2 . The following comments are in order. The cusp anoma lous dimension has a branch cut for x lying on the negative real axis. The results given in Eq. (9) are valid for 0 < x < 1 and can be analytically continued to other regions according to this choice of branch cuts [13]. The leading nj term in Eq. (9) is in agreement with the known result [14]. We reported on the ^-dependent part of Eq. (9) in Ref. [15]. The expression for the coefficient cx is new. As a check of our result, we can consider Minkowskian angles and take the lightlike limit, x = e~e with 6 -*■ 00, of Eq. (9), where one expects the behavior [16]
720
„ „ ^ . 209 5n2 7 „ + CACFnf TF[ - — + — - - C 3
(7)
1 cusp —
rifTpCp,
(17)
Remarkably, this quantity is independent of to three loops. Comparing to Eq. (15), we see that this means that,
062006-3
PHYSICAL
PRL 114, 062006 (2015)
REVIEW
week ending
LETTERS
13 FEBRUARY 2015
r > =4(as, y) —— CFA\ -\— ^ [A3 + a 2] n 2 \n ) ^4 _ A2 +
+ 0 ( a 4).
+ 63]
(20)
The two loop terms agree with Ref. [15]. As a test of the three loop prediction, we take the antiparallel fines limit and obtain I-.
/
\0
Op(Xs
FIG. 2. 6 dependence of the cusp anomalous dimension £l(a. e~B) at one (solid), two (dashed), and three (dotted) loops.
+ 1— 1 Jt/
1-
c\
5 4 + T _ 64
+ 0(al)
(3)
e.g., all ftf-dependent terms in T^/sp are generated from lower loop terms when expanding K(as) in as. In Fig. 2, we plot the one, two, and three loop coefficients of Q in an expansion of a / i t , for Minkowskian angles 9, i.e., x = e~° for the range 9 e [0,4], and with the number of colors set to N — 3. Note that the rif dependence in QCD can be obtained from Eq. (15) and amounts to a rescaling of the coupling. At large 9, the one loop contribution displays the linear behavior of Eq. (16), while the two and three loop contributions go to a constant, as expected. In the smallangle region, we have
Q.(a,e~e) = CF
)\e2
+ i - ) 3— ( - - -
\ji
12 l
3
0(0*).
(18)
The observed rif independence of £2(a, x) leads us to conjecture that the latter quantity is universal in gauge theories, i.e., independent of the specific particle content of the theory. Assuming this conjecture leads to a number of nontrivial predictions, as we discuss presently. First, let us recall the known value for K in J\f = 4 super Yang-Mills theory (in the DR scheme) [20],
K_\f=4(as) — CF 11 n
(19)
Plugging this formula and the result for £2 given in Eq. (17) into Eq. (15) then gives the previously unknown three loop result for the cusp anomalous dimension for the Wilson loop operator of Eq. (1) in that theory,
+0(c5°),
( 21)
as expected from the direct calculation of the quarkantiquark potential [21], Second, the conjecture of the rif independence of £2 can be used to predict the form of the nonplanar rif corrections that can first appear at four loops. The latter involve quartic Casimir operators of SU(N), whose con tribution we abbreviate by C4 = daFbcddaFbcd/ N A = ( 1 8 6N2 + N4)/ (96N2) [with NA the number of the SU(N) generators] [22], Consider a term in r cusp(as, x) of the form rif(as/n ) 4g(x)CFC4/ 64, for some g(x). Assuming that £2 defined in Eq. (15) is independent of tif then implies g(x) = g0A ]. Moreover, we can determine g0 by comparing to the antiparallel lines limit. The expected relation to the known quark-antiquark potential computed (numerically) in Ref. [23] then yields g0 = -56.83(1). In conclusion, we presented the full three loop result for the cusp anomalous dimension in QCD. The latter allows us to predict the infrared divergent part of planar scattering amplitudes of massive particles in QCD to that order. Moreover, our result can be applied to reduce theoretical uncertainties both in describing the scale dependence of heavy meson form factors [1,2] and in computing cross sections of top-antitop pair production in electron-positron annihilation and in hadronic collisions [5,24] (for a recent review, see Ref. [25]). We observed that the result has a surprisingly simple dependence on the number of quark flavors rif, which led us to define a quantity £2, independent of nj to three loops. If the latter is the same in any gauge theory, it could be studied using powerful integrability techniques that have been developed in J\f — 4 super Yang-Mills theory; see Ref. [26] for more details. A. G.’s work was supported by RFBR Grant No. 12-0200106-a and by the Russian Ministry of Education and Science. J. M. H. is supported in part by the DOE Grant No. DE-SC0009988, the Marvin L. Goldberger fund, and the Munich Institute for Astro- and Particle Physics
062006-4
PRL 114, 062006 (2015)
PHYSICAL
REVIEW
(MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe.” G. P. K. is supported in part by the French National Agency for Research (ANR) under Contract No. BLANC-SIMI-4-2011 (Stronglnt). P. M. was supported in part by the DFG through the SFB/TR 9 “Computational Particle Physics” and by the EU net works LHCPhenoNet and HiggsTools under Contract No. PITN-GA-2010-264564 and PITN-GA-2012-316704. We thank J. Bliimlein and J. Ablinger for making the “HarmonicSums” computer code based on Ref. [27] avail able to us.
*
[email protected] [email protected] "Gregory. Korchemsky @cea.fr §peter.marquard @desy.de [1] G. P. Korchemsky and A. V. Radyushkin, Phys. Lett. B 171, 459 (1986); Phys. Lett. B 279, 359 (1992). [2] M. Neubert, Phys. Rep. 245,259 (1994); A. V. Manohar and M. B. Wise, Heavy Quark Physics, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, England, 2000), Vol. 10, p. 1; A. G. Grozin, Springer Tracts Mod. Phys. 201, 1 (2004). [3] A .M . Polyakov, Nucl. Phys. B164, 171 (1980); R. A. Brandt, F. Neri, and M. A. Sato, Phys. Rev. D 24, 879 (1981). [4] G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys. B283, 342 (1987). [5] N. Kidonakis, Phys. Rev. Lett. 102, 232003 (2009). [6] P. Nogueira, J. Comput. Phys. 105, 279 (1993); J.A .M . Vermaseren, arXiv:math-ph/0010025. [7] K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B192, 159 (1981); A. V. Smirnov, J. High Energy Phys. 10 (2008) 107; R. N. Lee, arXiv:1212.2685; P. Marquard and D. Seidel (unpublished). [8] J.M . Henn, Phys. Rev. Lett. 110, 251601 (2013). [9] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.l 14.062006^ for the explicit basis choice / , and the result for / in the e expansion, to the order required for this calculation. [10] A. G. Grozin, J. High Energy Phys. 03 (2000) 013; K. G. Chetyrkin and A. G. Grozin, Nucl. Phys. B666, 289 (2003). [11] E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A 15, 725 (2000); D. Maitre, Comput. Phys. Commun. 174, 222 (2006).
LETTERS
week ending 13 FEBRUARY 2015
[12] The function A5 (and Aj and A3) appeared previously in the result for the cusp anomalous dimension in M = 4 super Yang-Mills theory for a locally supersymmetric Wilson line operator [28]. [13] Note that rcusp(as,-x) is expected to be related to r cusp(« v.x) by crossing symmetry, i.e., the two functions should be equal, up to terms picked up by the analytic continuation. It turns out that all functions except B5 contain rational factors that are symmetric under x -*■ - x , and, therefore, the harmonic polylogarithms in these expressions can be written with argument 1 —x2, and positive indices only. In contrast, the factor x / ( l —x2) contained in Bs is antisymmetric under x -* - x , and so is the transcendental function multiplying it (up to terms coming from branch cuts). [14] M. Beneke and V. M. Braun, Nucl. Phys. B454, 253 (1995). [15] A. Grozin, J. M. Henn, G. P. Korchemsky, and P. Marquard, Proc. Sci., LL2014 (2014) 016. [16] G.P. Korchemsky, Mod. Phys. Lett. A 04, 1257 (1989); G. P. Korchemsky and G. Marchesini, Nucl. Phys. B406, 225 (1993). [17] A. Vogt, Phys. Lett. B 497, 228 (2001); C. F. Berger, Phys. Rev. D 66, 116002 (2002); S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B 688, 101 (2004). [18] W. Kilian, T. Mannel, and T. Ohl, Phys. Lett. B 304, 311 (1993). [19] M. Peter, Phys. Rev. Lett. 78,602 (1997); Y. Schroder, Phys. Lett. B 447, 321 (1999). [20] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko, and V. N. Velizhanin, Phys. Lett. B 595, 521 (2004); 632, 754(E) (2006). [21] M. Prausa and M. Steinhauser, Phys. Rev. D 88, 025029 (2013). [22] T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin, Phys. Lett. B 400, 379 (1997). [23] A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Lett. B 668, 293 (2008). [24] M. Czakon, A. Mitov, and G. F. Sterman, Phys. Rev. D 80, 074017 (2009). [25] N. Kidonakis, DESY Report No. DESY-PROC-2013-03, 2014. [26] D. Correa, J. Maldacena, and A. Sever, J. High Energy Phys. 08 (2012) 134; N. Drukker, J. High Energy Phys. 10 (2013) 135. [27] J. Ablinger, arXiv:1011.1176; arXiv:1305.0687; J. Ablinger, J. Bliimlein, and C. Schneider, J. Math. Phys. (N.Y.) 52, 102301 (2011); J. Math. Phys. (N.Y.) 54, 082301 (2013). [28] D. Correa, J. Henn, J. Maldacena, and A. Sever, J. High Energy Phys. 05 (2012) 098.
062006-5
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PRL 114, 062006 (2015)
REVIEW
week ending 13 FEBRUARY 2015
LETTERS
Three Loop Cusp Anomalous Dimension in QCD $
Andrey Grozin Budker Institute o f Nuclear Physics SB RAS, Novosibirsk 630090, Russia and Novosibirsk State University, Novosibirsk 630090, Russia Johannes M. Henn' Institute for Advanced Study, Princeton, New Jersy 08540, USA Gregory P. Korchemsky* Institut de Physique Theorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France Peter Marquard§ Deutsches Elektronen-Synchrotron, DESY, Platanenallee 6, D15738 Zeuthen, Germany (Received 12 September 2014; published 13 February 2015) We present the full analytic result for the three loop angle-dependent cusp anomalous dimension in QCD. With this result, infrared divergences of planar scattering processes with massive particles can be predicted to that order. Moreover, we define a closely related quantity in terms of an effective coupling defined by the lightlike cusp anomalous dimension. We find evidence that this quantity is universal for any gauge theory and use this observation to predict the nonplanar ^-dependent terms of the four loop cusp anomalous dimension. DOI: 10.1103/PhysRevLett.ll4.062006
PACS numbers: 12.39.Hg, 11.15.-q
Understanding the structure of soft and collinear diver gences is of great theoretical interest in quantum field theory. It is also relevant for phenomenological applications such as the production of heavy particles at the LHC, where effects from soft gluon radiation need to be resummed in order to improve theoretical predictions. It is well known that the infrared (or long-distance) asymptotics of scattering amplitudes is described by correlation functions of Wilson lines pointing along the momenta of the scattered particles [1,2]. The latter satisfy evolution equations with the corresponding anomalous dimension being, in general, a matrix in color space. In the planar limit, this matrix is expressed in terms of the twoline cusp anomalous dimension [3]. The two loop result for this fundamental quantity has been known for more than 25 years [4]; see, also, Ref. [5], Here we report on the full result for the cusp anomalous dimension in QCD at three loops. To compute the cusp anomalous dimension, we consider the vacuum expectation value of the Wilson line operator W
N
,
= VyV2,
0031 -9 0 0 7 /1 5 /1 14(6)/062006(5)
x = e1^,
2cos = x + l / x .
(3)
In Euclidean space | jc| = 1, whereas for Minkowskian angles (p = id (with 6 real), the variable x can take arbitrary
( 1)
with A ^ x ) = A °(x )T a and T a being the generators of the fundamental representation of the SU (N ) gauge group. Here, the integration contour C is formed by two segments along directions v\ and (with v\ = v\ — Y), with (Euclidean space) cusp angle tp, COS (p
cf. Fig. 1. Perturbative corrections to the Wilson loop (1) contain both ultraviolet (cusp) and infrared divergences. We employ dimensional regularization with D = 4 - 2e to regularize the former and introduce an infrared cutoff using the heavy quark effective theory framework. The cusp anomalous dimension Tcusp is extracted as the residue at the simple pole l / e in the corresponding renormalization factor. It depends on the cusp angle 1
* {1) = Cp,
K® = CACF
(6)
The previously known one and two loop [4] results can be written as p (l)
_ C A
week ending 13 FEBRUARY 2015
LETTERS
'67
/r2\
5
36
12 )
9
'245
67 n2
96
216
+ C2FrifTp( £3 liusp
—
67 5 V — CfC4 - - C FTFn f \ A x.
x cFcA[a 3 + a 2] +
( 8)
At three loops, we find "cusp — c\CFC\ + c2Cf (TFnf y + CjCpT Fnf + c^CFCAT Ffij,
(9)
with 1
c\ — ^ [^5 +
a 4 + b 5 + b 3] +
67 29 — a 3 + Yg^2
'245 11 \ ~ > + 2 4 CV A l ’ 1 =~ 2 J A l ’
c4
( 55\ ^ 1, C3 = ( & - 4 8 )
209 n 5 r~ ~ , 1 /' - 9 [a 3 + a 2] - - ( % + l 6 ,K
( 10)
(ii)
- cusp (as,x)x= K
( a s) log(l/jc) + O(x0),
55
1
48/
27
Cp(nf TF)2,
(14)
rcusp).
Q.(a,x)—r cusp(as,x),
a ~ n / C FK(as),
(15)
where Tcusp and K(as) are evaluated in the same scheme (and for the same theory). By construction, Q has the universal limit S2(a,x)JC= 0- C f-l og(l /x) + O(x0),
n
(16)
as one can easily verify by comparing to Eq. (13). Using the results up to three loops given in Eqs. (7)-(9) and (14), and expanding both sides of the first relation in Eq. (15) to third order in as, we find
f a \ 2 CACF
£2(m, x) — —CpAx T [ — (« )
+
a \ 3CpC2A
A A ~A A3 + A2 + K—2 A t
2 f
A5 + A4 —A 2 + B5 +
nA ~ n2 ~ n2 ~ + y A 3 + T A2 - — Aj + 0 ( a 4).
(13)
with K(as) being the lightlike cusp anomalous dimension. To three loops, it is given by [17]
24'
where K(as) = ^2m>i{ocs/n)mK^m\ We found perfect agreement for all terms. Finally, if the conformal symmetry of (massless) QCD were not broken, one would expect that the cusp anomalous dimension should be related in the antiparallel lines limit (p — n - 8, 8 -* 0, to the quark-antiquark potential [18] (at one loop order lower compared to Starting from Eq. (9), we indeed find perfect agreement with the result quoted in the second of Ref. [19], up to conformal symmetry breaking terms proportional to the QCD /I function. Our result for the cusp anomalous dimension is valid in the MS (dimensional regularization) scheme. Going to the DR (dimensional reduction) scheme amounts to a finite renormalization of the coupling constant. We can introduce a quantity Q. which is the same in both schemes by switching from as to an “effective cou pling” a,
( 12)
Here, CF = (N2 - 1 ) /(22V) and CA = N are the quadratic Casimir operators of the SU(N) gauge group in the fundamental and adjoint representation, respectively, rif is the number of quark flavors, and TF = 1/ 2 . The following comments are in order. The cusp anoma lous dimension has a branch cut for x lying on the negative real axis. The results given in Eq. (9) are valid for 0 < x < 1 and can be analytically continued to other regions according to this choice of branch cuts [13]. The leading nj term in Eq. (9) is in agreement with the known result [14]. We reported on the ^-dependent part of Eq. (9) in Ref. [15]. The expression for the coefficient cx is new. As a check of our result, we can consider Minkowskian angles and take the lightlike limit, x = e~e with 6 -*■ 00, of Eq. (9), where one expects the behavior [16]
720
„ „ ^ . 209 5n2 7 „ + CACFnf TF[ - — + — - - C 3
(7)
1 cusp —
rifTpCp,
(17)
Remarkably, this quantity is independent of to three loops. Comparing to Eq. (15), we see that this means that,
062006-3
PHYSICAL
PRL 114, 062006 (2015)
REVIEW
week ending
LETTERS
13 FEBRUARY 2015
r > =4(as, y) —— CFA\ -\— ^ [A3 + a 2] n 2 \n ) ^4 _ A2 +
+ 0 ( a 4).
+ 63]
(20)
The two loop terms agree with Ref. [15]. As a test of the three loop prediction, we take the antiparallel fines limit and obtain I-.
/
\0
Op(Xs
FIG. 2. 6 dependence of the cusp anomalous dimension £l(a. e~B) at one (solid), two (dashed), and three (dotted) loops.
+ 1— 1 Jt/
1-
c\
5 4 + T _ 64
+ 0(al)
(3)
e.g., all ftf-dependent terms in T^/sp are generated from lower loop terms when expanding K(as) in as. In Fig. 2, we plot the one, two, and three loop coefficients of Q in an expansion of a / i t , for Minkowskian angles 9, i.e., x = e~° for the range 9 e [0,4], and with the number of colors set to N — 3. Note that the rif dependence in QCD can be obtained from Eq. (15) and amounts to a rescaling of the coupling. At large 9, the one loop contribution displays the linear behavior of Eq. (16), while the two and three loop contributions go to a constant, as expected. In the smallangle region, we have
Q.(a,e~e) = CF
)\e2
+ i - ) 3— ( - - -
\ji
12 l
3
0(0*).
(18)
The observed rif independence of £2(a, x) leads us to conjecture that the latter quantity is universal in gauge theories, i.e., independent of the specific particle content of the theory. Assuming this conjecture leads to a number of nontrivial predictions, as we discuss presently. First, let us recall the known value for K in J\f = 4 super Yang-Mills theory (in the DR scheme) [20],
K_\f=4(as) — CF 11 n
(19)
Plugging this formula and the result for £2 given in Eq. (17) into Eq. (15) then gives the previously unknown three loop result for the cusp anomalous dimension for the Wilson loop operator of Eq. (1) in that theory,
+0(c5°),
( 21)
as expected from the direct calculation of the quarkantiquark potential [21], Second, the conjecture of the rif independence of £2 can be used to predict the form of the nonplanar rif corrections that can first appear at four loops. The latter involve quartic Casimir operators of SU(N), whose con tribution we abbreviate by C4 = daFbcddaFbcd/ N A = ( 1 8 6N2 + N4)/ (96N2) [with NA the number of the SU(N) generators] [22], Consider a term in r cusp(as, x) of the form rif(as/n ) 4g(x)CFC4/ 64, for some g(x). Assuming that £2 defined in Eq. (15) is independent of tif then implies g(x) = g0A ]. Moreover, we can determine g0 by comparing to the antiparallel lines limit. The expected relation to the known quark-antiquark potential computed (numerically) in Ref. [23] then yields g0 = -56.83(1). In conclusion, we presented the full three loop result for the cusp anomalous dimension in QCD. The latter allows us to predict the infrared divergent part of planar scattering amplitudes of massive particles in QCD to that order. Moreover, our result can be applied to reduce theoretical uncertainties both in describing the scale dependence of heavy meson form factors [1,2] and in computing cross sections of top-antitop pair production in electron-positron annihilation and in hadronic collisions [5,24] (for a recent review, see Ref. [25]). We observed that the result has a surprisingly simple dependence on the number of quark flavors rif, which led us to define a quantity £2, independent of nj to three loops. If the latter is the same in any gauge theory, it could be studied using powerful integrability techniques that have been developed in J\f — 4 super Yang-Mills theory; see Ref. [26] for more details. A. G.’s work was supported by RFBR Grant No. 12-0200106-a and by the Russian Ministry of Education and Science. J. M. H. is supported in part by the DOE Grant No. DE-SC0009988, the Marvin L. Goldberger fund, and the Munich Institute for Astro- and Particle Physics
062006-4
PRL 114, 062006 (2015)
PHYSICAL
REVIEW
(MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe.” G. P. K. is supported in part by the French National Agency for Research (ANR) under Contract No. BLANC-SIMI-4-2011 (Stronglnt). P. M. was supported in part by the DFG through the SFB/TR 9 “Computational Particle Physics” and by the EU net works LHCPhenoNet and HiggsTools under Contract No. PITN-GA-2010-264564 and PITN-GA-2012-316704. We thank J. Bliimlein and J. Ablinger for making the “HarmonicSums” computer code based on Ref. [27] avail able to us.
*
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